Proof Methods Showcase (MAT 3100 assignment template)
Autor
The BFG
Letzte Aktualisierung
3 years ago
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Creative Commons CC BY 4.0
Abstrakt
MAT 3100 assignment template
MAT 3100 assignment template
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsthm,amssymb,amsmath}
\newtheorem*{theorem}{Theorem}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\title{Proof Methods Showcase}
\author{Me} % Replace Me with your name!
\date{\today}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Direct Proof}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Contrapositive}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Contradiction}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method. (Reviewing Section 3.6 can help.)
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof by Mathematical Induction}
Replace this text with a few sentences describing the process for this type of proof and when you might use this method.
\begin{theorem}
Replace this text with the theorem statement.
\end{theorem}
\begin{proof}
Write your proof here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Some LaTeX Commands}
Here are some example sentences using LaTeX commands:\\
If $a\equiv 2\pmod{3}$, then $a^2\equiv 1\pmod 3$.\\
If $x$ and $y$ are positive real numbers, the arithmetic mean is $\dfrac{x+y}{2}$ and the geometric mean is $\sqrt{xy}$.\\
The union of two sets is $A\cup B$ and the intersection of two sets is $A\cap B$.\\
Let $(x,y)\in A\times B$.\\
Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^{2019}$.\\
In set-builder notation, the set of all odd integers is $\{2k+1\mid k\in\mathbb{Z}\}$.\\
Suppose that
\[1+3+5+\cdots+(2k-1) = k^2.\]
Note that if $a=2k$, then
\begin{align*}
a^2+3a+5 &= (2k)^2+3(2k)+5 \\
&= 4k^2+6k+4+1 \\
&= 2(2k^2+3k+2)+1.
\end{align*}
If $g\circ f$ is surjective, then $g$ is surjective.\\
Note that $\NN\subseteq \ZZ\subseteq \QQ\subseteq\RR\subseteq\CC$.
\end{document}